# Hybrid Systems

**Hybrid systems** combine time-based signals with event sequences. To effectively
combine them, the events in the event sequence have to be associated with a
time. Thus, we replace the event sequence

*x*: *Naturals*_{0} → *Symbols*

with

*x*: *T* → *Symbols*

where *T* is either *Naturals*_{0}, *Integers*, *Reals*+,
or *Reals.* For most *t* in *T*, *x*(*t*) = *absent*,
the stuttering element in *Symbols*. When *x*(*t*) is not *absent*,
then we interpret *t* to mean the time at which the event *x*(*t*)
occurs.

Now we can define systems that mix time-based signals with event sequences.